Boolean cardinality constraints (CCs) state that at most (at least, or exactly) k out of n propositional literals can be true. We propose a new, arc-consistent, easy to implement and efficient encoding of CCs based on a new class of selection networks. Several comparator networks have been recently proposed for encoding CCs and experiments have proved their efficiency (Abío et al. 2013, Asín et al. Constraints 12(2): 195–221, 2011, Codish and Zazon-Ivry 2010, Eén and Sörensson Boolean Modeling and Computation 2: 1–26, 2006). In our construction we use the idea of the multiway merge sorting networks by Lee and Batcher (1995) that generalizes the technique of odd-even sorting ones by merging simultaneously more than two subsequences. The new selection network merges 4 subsequences in that way. Based on this construction, we can encode more efficiently comparators in the combine phase of the network: instead of encoding each comparator separately by 3 clauses and 2 additional variables, we propose an encoding scheme that requires 5 clauses and 2 variables on average for each pair of comparators. We also extend the model of comparator networks so that the basic components are not only comparators (2-sorters) but more general m-sorters, for m ∈ {2, 3, 4}, that can also be encoded efficiently. We show that with small overhead (regarding implementation complexity) we can achieve a significant improvement in SAT-solver runtime for many test cases. We prove that the new encoding is competitive to the other state-of-the-art encodings.

中文翻译：

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使用多路合并选择网络编码基数约束

布尔基数约束（CC）指出，n个命题文字中最多（至少或完全）k个可以为真。我们基于一类新的选择网络，提出了一种新的，弧线一致，易于实现且高效的CC编码。最近已经提出了几种比较器网络来编码CC，并且实验证明了它们的效率（Abío等人，2013，Asín等人，Constraints 12（2）：195–221，2011，Codish和Zazon-Ivry 2010，Eén和SörenssonBoolean建模与计算2：2006年1月26日）。在我们的构造中，我们使用Lee和Batcher（1995）提出的多路合并排序网络的思想，该思想通过同时合并两个以上子序列来推广奇偶排序技术。新的选择网络以这种方式合并了4个子序列。基于此构造，我们可以在网络的合并阶段对比较器进行更有效的编码：我们不建议通过3个子句和2个其他变量分别对每个比较器进行编码，而是建议一种编码方案，每对平均需要5个子句和2个变量比较器。我们还延伸比较器网络的模型，以使基本部件是不仅比较（2-分拣机），但更一般的米-sorters，为米∈{2，3，4}，也可以有效地对其进行编码。我们证明，以较少的开销（关于实现的复杂性），我们可以在许多测试案例中实现SAT求解器运行时的显着改善。我们证明了新的编码与其他最新的编码相比具有竞争力。